The article by
J.J. Monaghan,
The Heaviside-Feynman expression for the fields of an accelerated dipole,
J. Phys. A: 1, 112, 1968
extends the Heaviside-Feynman expressions to the case of a point object with
electric and magnetic dipole moments.

Electromagnetic Field Equation

Coulomb's Law for the electric field holds exactly only for static charges.
It implies instantaneous action at a distance and does not take into account
the finite speed of light.

The inverse square law of gravity was discovered by Newton, but he was never
comfortable with it.

That one body may act upon another at a distance through a vacuum without the
mediation of anything else, by and through which their action and force may be
conveyed from one another, is to me so great an absurdity that, I believe, no
man who has in philosophic matters a competent faculty of thinking could ever
fall into it.
—
Isaac Newton, third letter to Richard Bentley, 1692.

The fundamental problem of classical electrodynamics is to derive the exact
electromagnetic field due to a point charge moving with arbitrary acceleration.
The electromagnetic field due to a system of charges with given accelerations
can then be derived by linear superposition.

The concept of a potential field was first introduced by Lagrange (in a 1773
article on the secular equation of the Moon) to
explain gravitational action at a distance.
Coulomb's Law is obtained by solving Poisson's equation for the
electrostatic potential.

The fully relativistic analog of Poisson's equation in electrostatics is

Relativistic Index Notation

Subscript 4-vector indices denote covariant components.
I write 4-vectors with the time component last.
The contravariant components of the space-time point 4-vector are written

The covariant time component has the opposite sign

Einstein's summation convention is used for repeated indices, one upper and
the other lower:

The covariant gradient is

Note the superscript in the denominator!

The d'Alembertian wave operator is

The contravariant and covariant photon field components are

with SI units volts.

The contravariant and covariant current density components are

with SI units .

The decoupled form of the electromagnetic wave equation
assumes the Lorenz gauge fixing condition

Note that the every equation has exactly the same form as the nonrelativistic
equations in Jackson Chapter 6.

Note also that I use different units and conventions compared with Jackson
Chapters 11-16. The time component is written following the space components,
and the metric tensor in Minkowski space

which is used to raise and lower indices

has the opposite sign from Jackson. SI units are used consistently in my
notes. These are just conventions.
Choose the convention you are most comfortable with and use it consistently.

Causal Green's Function for the Field Equation

A Green's function is a particular solution of a linear partial differential
equation with a point source and a choice of boundary conditions.

In relativistic physics information can only be transmitted forward in time
and it cannot travel faster than the speed of light.
This requirement is called causality.

The causal Green's function for the d'Alembertian operator is a
particular solution of

The right hand side represents a unit source at the origin of spacetime.
Causal boundary conditions require that the solution is zero at every
spacetime point outside of the forward light cone with vertex at the
origin of spacetime.

This solution is singular for Fourier modes that propagate with the
speed of light!

Singularities in mathematics need to be defined using a limiting process.
Different limiting procedures will in general give different solutions.
The physical solution that we require is selected by specifying causal
boundary conditions.

Singularities in differential equations can be defined by analytic continuation
in one or more integration variables.
Consider the integral

The integral will vanish exponentially as if
is given a small positive imaginary
part.
This is the behavior we need for forward propagation in time.
In fact, this procedure gives causal propagation.

For any the integrand vanishes exponentially on the infinite
semicircle with .
The integration contour can be closed in the upper half plane, which
excludes the two poles.

For the integration contour is closed in the lower half plane,
which includes the two poles, and can be evaluated using the method of
residues.

The causal Green's function is

The Green's function for an instantaneous source at any other spacetime
point is

This satisfies

Fields of an Accelerating Point Charge

The causal Green's function solves the electromagnetic wave equation

with physically correct causal boundary conditions.

The observer measures the photon field at spacetime point .
The effect is calculated by integrating the current density
over all spacetime points ,
in the past and future and at every location in space.

Relativisitic Velocity

The mass and electric charge of a particle are constants that do not
depend on the reference frame of the observer.
Mass and charge are relativistically invariant.

The spacetime position of the particle is a 4-vector.
A spacetime displacement is also a 4-vector

The time interval is not and invariant: it depends on the frame of
the observer.
This means that the nonrelativistic velocity

is not the space part of a 4-vector, so it does not transform in a simple
way.

To obtain the relativistic 3-vector velocity we must divide the 3-vector
displacement by an invariant time interval called the proper time interval

Proper time is just the time measured in the rest frame of the particle.

Relativistic Current Density

The electric charge of a particle is a constant and has the same
value in any reference frame.

Charge density at a point in space is defined by finding the
net charge
inside a small spatial volume containing the point
and taking the limit

This is not Lorentz invariant because is the same in every
reference frame but is not.

The spatial current density is

where is the velocity of in the limit
.

We saw in defining four velocity that is not the spatial
part of the relativistic velocity.
Rather amazingly, the product of with (which is not
an invariant, is the spatial part of a relativistic 4-vector!

To show this, consider a small time interval during which the
charge in moves .
The quantity

is a relativistic 4-vector, and

is relativistically invariant: under a Lorentz transformation
contracts in the boost direction, and dilates by the same factor!

For a system of point particles with charges and positions

The relativistic 4-vector current density of charge is

This can be put in manifestly relativistic form using the relativistic
4-velocity and proper time of the charge .

The time in the observer's frame and the proper time in the
particle's rest frame both increase continuously but at different rates.
Let be the function the particle would use to compute the
time in the observer's frame.

To prove the last equality use the inverse function

which also implies

This gives the manifestly relativistic formula

Liénard-Wiechert Potentials

The causal Green's function solution for charge is

The subscript means that the proper time is
evaluated using the causality constraint

To solve this constraint find the instant on the world line of the
charge such that the forward light cone at this instant
intersects the spacetime position at which the
observer measures the field.

Heaviside-Feynman Formulas

The electric field formula in Feynman's Lectures on Physics Chapter 28:

The first term is the Coulomb field evaluated with the distance to the charge
determined by causality.

The second term is a correction to the Coulomb field which falls off more
rapidly than the square of the distance.

The third term represents radiation, and falls off inversely as the
distance.

Radiation Field of a Moving Charge

J.J. Thomson gave a good physical derivation of the radiation field in
his 1907 book "Mathematical Theory of Electricity and Magnetism".
Figure 4.6 below from Misner-Thorne-Wheeler illustrates his explanation.

Assignment 2 is due February 5 on UBlearns

Please format your submission as a single PDF file according to the
instructions in the syllabus and
submit it before 11:59 pm on Sunday, February 5.

Problem 1:
Starting from the Liénard-Wiechert potential

derive Feynman's expression

Problem 2:
Understand the definitions and units used in Jackson's expression

and show that it is equivalent to Feynman's expression in Problem 1.

Problem 3:
Starting from the Liénard-Wiechert potential in Problem 1, derive
the Heaviside expression for the magnetic field given in Jackson

Did you need to change units or make any non-relativistic approximations?

Problem 4:
Understand and summarize the derivation of the dipole potentials in section
3 of the article by J.J. Monaghan,
The Heaviside-Feynman expression for the fields of an accelerated dipole,
J. Phys. A: 1, 112, 1968.
Write Monaghan's expressions for the potential using SI units and relativistic
4-vector notation as in the Liénard-Wiechert potential expression
in Problem 1.